Precalc
posted by Grace .
The figure shows a 10 x 10 square centered at the origin. The darkershaded square is the image of this square, dilated by a factor of 0.8 and rotated about the origin by exactly the number of degrees, , so that its corners touch the sides of the parent square. The corners of the rotated square divide the sides of the 10 x 10 square into segments of length a and b.
Use the fact that a and b add up to 10 to find the value of theta.

Not without the figure. I cannot visualize the situation from your verbal description. Perhaps another teacher can.

You end up with a rotated square of side 8, surrounded by 4 right triangles of sides x and 10x.
x^2 + (10x)^2 = 64
x^2 + 100  20x + x^2 = 64
2x^2  20x + 36 = 0
x^2  10x + 18 = 0
x = 5 ± √7 = 2.35 or 7.65
if we let x be the smaller value, then
tanθ = 2.35/7.65 = 0.307
θ = 17° 
Thank you so so so much! I've been stuck on this for hours!